Stone Duality for Stable Functions

The problem of finding an algebraic structure for stable open subsets of a suitable domain has been recently raised by several authors. Specifically, it could be useful to have, in the stable case, a notion similar to the one of “frame”, in order to develop something similar to pointless topology, that is an algebraic insight of spaces. In domain theory, an interesting application of frame theory is the logic of domains developped by Abramsky (see [Ab]) in the topological case. The idea of constructing a logic for “stable properties” has been originated by Zhang (see [Zha]), who got very interesting results about “stable opens” of dI domains. However, his notions remain concrete, and it is not clear whether they give rise to a duality. In that sense, they lack the “localic” properties which justify the canonicity of Abramsky’s approach. We introduce the notion of S-Structure as the structure of the algebra of stable open sets intended to correspond to the concept of frame in the stable case. These S-structures have properties which are very similar to the ones of frames, from the point of view of duality. So we may hope to achieve a logic of domains as natural as Abramsky’s one, but expressing properties of programs which are not captured by the continuous approach. In this paper, we give the fundamentals of S-structures theory. We prove first general duality results which do not involve any domain theoretical assumtion about spaces. Actually we introduce the S-spaces which play wrt S-structures the same role as topological spaces wrt frames. These duality results can be specialized to the case of domains, and then we obtain a result similar to known Scotttopological duality in domain theory. The corresponding notion of domain widely subsumes the usual dI domains to which stabilty theory is usually restricted. Indeed these domains are the most general ones where stability makes sense. By duality method it is rather simple to treat function spaces and we retrieve cartesian closedness of the category of L-domains and stable functions which has been recently proved by P. Taylor.